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<h3 style="text-align:center;">Number of states of an ideal gas: Finite size effects</h3>

<p class="header_title">Introduction</p>

<p>As discussed in Pathria and many other texts, the number of states for one particle in a two-dimensional box is given by the values of n<sub>x</sub> and n<sub>y</sub> that satisfy the condition</p>
<p class="center">
<img src="condition2.jpg" alt="" align="middle" >
</p>
<p>where the quantum numbers n<sub>x</sub> and n<sub>y</sub> are nonzero positive integers. R is related to the energy of the system by</p>
<p class="center">
<img src="R2.jpg" alt="" align="middle" >
</p>
<p>where L is the linear dimension of the box, m is the mass of the particle, and h is Planck's constant.</p>

<p class="center">
<img src="states.jpg" alt="" width="367" height="357" align="middle" >
</p>

<p>&nbsp; &nbsp;&nbsp;&nbsp;In the semiclassical limit where E is large, the number of states with energy less than or equal to E is given by the area of the positive quadrant of a circle of radius r:</p>
<p class="center">
<img src="number2.jpg" alt="" align="middle" >
</p>
<p>However, the number of states for finite values of R (and E) is different than this asymptotic expression (see the figure).</p>

<center>
<applet
 code="org.opensourcephysics.davidson.applets.ApplicationApplet.class"
 archive="./stp.jar" codebase="../" align="top" height="40"
 hspace="0" vspace="0" width="150"> <param name="target"
 value="org.opensourcephysics.stp.numOfStates.NumOfStatesApp"> <param name="title"
 value="Applet"> <param name="singleapp" value="true">
</applet>
</center>

<p>&nbsp;&nbsp;&nbsp;&nbsp;The applet/application shows the asymptotic expression and the actual number of states. The latter is given by</p>
<p class="center">
<img src="gammad.jpg" alt="" align="middle" >
</p>
<p>where d is the spatial dimension of the box and</p>
<p class="center">
<img src="sumd.jpg" alt="" align="middle" >
</p>

<p class="header_title">Problems</p>

<ol>

<li>Derive the various formulas that have been discussed.</li>

<li>Does the asymptotic expression for &#915; under or over estimate the actual number of states? (The program works for dimension d = 1,2, and 3.)</li>

<li>What is the minimum value of R such that the difference between the actual and the asymptotic expression for the number of states is less than one percent?</li>

</ol>

<p class="header_title">Reference</p>

<ul>
<li>R. K. Pathria, Statistical Mechanics, second edition, Butterworth-Heinemann (1996), pp. 18&#8211;19.</li>

</ul>

<p class="header_title">Java Classes</p>

<ul>

<li>NumOfStatesApp</li>
</ul>

<p class = "small">Updated 27 February 2007.</p>

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